Coulomb T matrix "half-off-shell" at zero momentum, and for large l.

Abstract

Recently Talukdar et al. (Phys. Rev. A 29, 1865 (1984)) found that the so-called ''half-off-shell'' Coulomb T matrix T/sub C/l(p,k;k/sup 2/) were to be (i) singular at p = 0 for l>0, and (ii) increasing for increasing l. We point out that both these results are incorrect: (i) From the simple closed formula t/sub C/l(p,k;k/sup 2/) = (2..gamma../..pi..p) exp(3..pi gamma../2)Q/sub l//sup i/..gamma..(p/sup 2/+k/sup 2/ )/2pk), 0..0, precisely as the Coulomb-potential matrix element and the off-shell Coulomb T matrix. (..gamma.. is Sommerfeld's parameter and Q/sub l//sup i/..gamma.. is Legendre's function of the second kind.) (ii) For all pnot =k, t/sub C/l(p,k;k/sup 2/) tends exponentially to zero as l tends to infinity: as l/sup -1/2/ exp(-lVertical Barln(p/k)Vertical Bar).